Lpwan communication protocol design with turbo codes

ABSTRACT

A method and a decoder for receiving a message encoded in Turbo Codes and modulated for transmission as an analog signal includes: (a) demodulating the analog signal to recover the Turbo Codes; and (b) decoding the Turbo Codes to recover the message using an iterative Turbo Code decoder, wherein the decoding includes performing an error detection after a predetermined number of iterations of the Turbo Code decoder to determine whether or not an error has occurred during the transmission. The predetermined number of iterations may be, for example, two. Depending on the result of the error detection, the decoding may stop, a request for retransmission of the message may be sent, or further iterations of decoding in the Turbo Code decoder may be carried out.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims priority of U.S. provisional applicationSer. No. 62/749,793, filed on Oct. 23, 2018, entitled “LPWANCommunication Protocol Design With Turbo Codes.”

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to communication protocols. In particular,the present invention relates to communication protocols that aresuitable for use in wireless communication applications, such asLow-Power Wide-Area Networks (LPWANs).

2. Discussion of the Related Art

FIG. 1 shows model 100 of a communication system. As shown in FIG. 1,binary data 101 is encoded into a sequence of binary code words at step102. Examples of encoding include error detecting codes, errorcorrection codes and turbo codes. In the detailed description below, anencoding scheme that uses turbo codes, or any of several selected typesof error detecting codes, illustrates the present invention. The encodedsequence of binary code words is modulated at step 103 onto a carriersignal for transmission at step 104. One example of a modulation schememay be code division multi-access (CDMA), combined with eitherquadrature phase-shift keying (QPSK) modulation or binary phase-shiftkeying (BPSK) modulation. The modulated signal may be transmitted over anoisy communication channel and received at a receiver at step 106.

At step 107, the receiver demodulates the signal received from thecommunication channel to recover the encoded signal, which is thendecoded at step 108 to recover binary data 109. An error that occursalong the signal path represented by steps 102-107 may cause binary data109 and binary data 101 to be different.

In communication model 100, noise in the communication channel betweentransmission at step 106 and reception at step 107 may be modeled by aGaussian noise model (“Gaussian channel”). In some instances, modulationstep 105 and demodulation step 107 may be lumped for the purpose ofanalysis into the communication channel (“binary channel”).

In this detailed description, embodiments of the present invention aredescribed using a CDMA modulation scheme, which may be characterized bythe carrier frequency f₀ and the number of chips (i.e., bits) N in thecode assigned to a signal source. The noise in the Gaussian channelunder a CDMA modulation scheme may be characterized by a signal-to-noiseratio (SNR) defined as:

${SNR} = {10^{\frac{\varphi}{10}} \times \frac{N}{f_{0}}}$

where ϕ represents a modulation scheme-independent noise measuredexpressed in dB Hz. The Gaussian channel has a channel capacity C_(g)defined as:

$C_{g} = {\min\limits_{N}{C_{s}\left( {N,\varphi} \right)}}$

where C_(s) (N, ϕ) is defined as:

${C_{S}\left( {N,\varphi} \right)} = {\frac{f_{0}}{2N}{\log \left( {1 + {10^{\frac{\varphi}{10}} \times \frac{N}{f_{0}}}} \right)}}$

For a sufficiently small

${10^{\frac{\varphi}{10}} \times \frac{N}{f_{0}}},{C_{g} \approx {\frac{1}{2\ln \; 2}10^{\frac{\varphi}{10}}}}$

Similarly, the binary channel has a capacity C_(B) defined as:

$C_{B} = {\min\limits_{N}{\frac{f_{0}}{N}{C_{b}\left( {N,\varphi} \right)}}}$

where C_(b) (N, ϕ) is the greatest coding rate achieved in an optimalcoding scheme, defined as:

C _(b)=1+p logp+(1−p)log(1−p),

where p is the probability that a bit in a transmitted code word isflipped, given by:

$p = {\frac{1}{2}\left( {1 - {{erf}\left( \frac{1}{\sigma \sqrt{2}} \right)}} \right)}$with${{{erf}(x)} = {\frac{1}{\sqrt{\pi}}{\int_{- x}^{x}{e^{- t^{2}}{dt}}}}},{and}$$\sigma = {\frac{1}{\sqrt{SNR}} = 10^{{\frac{- \varphi}{20} \times \frac{N}{f_{0}}}\;}}$

The information transmit rate and the decoding energy are two figures ofmerit that are often used in the analysis of a communication channel Itis desirable to have a high information transmit rate, while keeping thedecoding energy low.

Suppose an encoded message frame of length K, resulting from encoding rKbits of information (r being the “information rate,” typically less than1.0) is provided for transmission. The additional bits under theencoding scheme enable error detection, error correction, or both.Suppose further that the message frame is transmitted using a CDMAmodulation scheme with a channel coding rate η (i.e., the K-bit messageframe is encoded into a K/η-bit blocks for transmission), and thecommunication channel has an error rate P_(er) per block, the effectiveinformation rate R is given by:

$R = \frac{\eta \; {f_{0}\left( {1 - P_{er}} \right)}r}{N}$

The decoding energy per information bit is given by:

$E = {\eta \; {E_{0}(\eta)}\frac{f_{0}}{N}}$

for small P_(er), where E₀ (η) is the computational energy per inputbit.

The “Turbo code” encoding scheme is illustrated by Turbo Code encoder200 of FIG. 2a . As shown in FIG. 2a , Turbo Code encoder 200 receivesan input sequence {d_(k)} to provide an output sequence {X_(k), Y_(k)},where X_(k) reproduces sequence {d_(k)} and Y_(k) is the concatenationof two subsequences, Y_(1k) and Y_(2k). The output sequence is thentransmitted. In FIG. 2a , subsequences Y_(1k) and Y_(2k) are recursivesystematic convolutions (RSC) codes. In the example of FIG. 2a , RSCencoders 201 a and 201 b each include 4 unit-delay shift registers andboth implement the finite state machine:

${h(x)} = \frac{1 + {D^{4}x}}{1 + {Dx} + {D^{2}x} + {D^{3}x} + {D^{4}x}}$

Delay line 202 delays sequence {d_(k)} by 5 bits, so that both RSCencoders 201 a and 201 b operate on the same set of input bits.Interleaver 203 permutates the delayed sequence at RSC encoder 201 bbefore coding. Permutating the input sequence provides higherperformance in the presence of burst noise in the communication channel.In this example, the coding rate is nominally ⅓, as the nominal codingrate is

$\frac{1}{\left( {1 + n} \right)},$

where n is the number of RSC codes. In some embodiments, a predeterminednumber of bits may be removed (“punctured”) from subsequences Y_(1k) andY_(2k), so as to achieve a higher coding rate. In some embodiments,additional RSC encoders may be provided for greater noise immunity.Turbo Codes have been shown to achieve a channel capacity that isclosest to the theoretical Shannon limit.

FIG. 2b shows schematically exemplary Turbo Code decoder 250 fordecoding turbo codes that are generated from Turbo Code encoder 200. Asshown in FIG. 2b , Turbo Code decoder 250 includes two decoders 251 aand 251 b, each of which may be a BCJR decoder, as known to those ofordinary skill in the art. Each decoder provides an output valueΛ(d_(k)), computed based on input bit x_(k), y_(k), and Λ(d′_(k)), wherex_(k) and y_(k) are corresponding received bits from transmittedsequence {X_(k), Y_(k)} and Λ(d′_(k)) is a previous computed outputvalue from decoding the same x_(k) and y_(k). In FIG. 2b , decoder 251 aoperates on x_(k) and y_(1k) and the preceding output value of decoder251 b from the previous iteration, and decoder 251 b operate on x_(k)and y_(1k) and the output value of decoder 251 a in the currentiteration. In the example of FIG. 2b , Λ(d_(k)) may be the loglikelihood ratio:

${\Lambda \left( d_{k} \right)} = {\log \; \frac{p\left( {d_{k} = 1} \right)}{p\left( {d_{k} = 0} \right)}}$

Using the BJCR algorithm, known to those of ordinary skill in the art,one of the probabilities p(d_(k)=1) and p(d_(k)=0) increases while theother decreases with the number of iterations. The number of iterationsnecessary to achieve a given error bit rate can be determinedempirically for a communication channel. After the requisite number ofiterations, the sign of Λ(d_(k)) can be used to determine if the decodedbit is ‘1’ (positive A(d_(k)) or ‘)0’ (negative Λ(d_(k))).

While the complexity of Turbo Code encoding grows linearly with both thecoding rate and the number of unit delay elements in the RSC codeencoder, the complexity of Turbo Code decoding grows exponentially withthe number of unit delay elements in the RSC code encoder. For thisreason, in practical applications, the number of unit delay elements inthe RSC code encoder seldom exceeds 4. Furthermore, for manyapplications, it is not uncommon to find that the number of requireddecoding iterations exceeds 10 to achieve an acceptable bit error rate,resulting in a long decoding latency. Therefore, it is long desired toprovide an encoding and decoding scheme that achieves the near-Shannoncapacity of Turbo codes, while having a shorter decoding latency.

SUMMARY OF THE INVENTION

According to one embodiment of the present invention, a decoder and amethod for receiving a message encoded in Turbo Codes and modulated fortransmission as an analog signal are provided. The method includes: (a)demodulating the analog signal to recover the Turbo Codes; and (b)decoding the Turbo Codes to recover the message using an iterative TurboCode decoder, wherein the decoding includes performing an errordetection after a predetermined number of iterations of the Turbo Codedecoder to determine whether or not an error has occurred during thetransmission. The predetermined number of iterations may be, forexample, two. When the error detection determines that an error haslikely not occurred during the transmission, the decoding may stop. Suchan error detection may include, for example, (i) computing the RSC codesfrom the decoded message portion of the Turbo Codes, and (ii) evaluatingthe computed RSC codes against the decoded RSC codes portion. When theerror detection determines that an error has occurred during thetransmission, a request for retransmission of the message may be sentor, alternatively, one or more further iterations of decoding in theTurbo Code decoder may be carried out.

According to one embodiment of the present invention, the errordetection step may include evaluating an indicator which provides aprobability that an error has occurred during the transmission. Theanalog signal may be transmitted at one of a plurality of analog values(e.g., 1.0 and −1.0). The error detection may involve evaluating aprobability P (Y_(i)-X_(i)|X_(i)), which is the probability ofreceiving, for the i-th bit of a Turbo Code block an analog value Y_(i),given a transmitted analog signal X_(i). In one implementation using abinary modulation scheme, in which the values X_(i) and X _(l) are theanalog representations of complementary binary values ‘1’ and ‘0’, onemay calculate an indicator of error occurrence during transmission usingthe expression:

$\frac{- 1}{L}{\sum\limits_{i = 1}^{L}{\log \; \frac{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)}{{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)} + {P\left( {Y_{i} - \overset{\_}{X_{i}}} \middle| \overset{\_}{X_{i}} \right)}}}}$

where L is a length in number of bits of the Turbo Code block.

Alternatively, or in addition to evaluating the indicator, the TurboCode block encodes an error detection code word (e.g., a cyclicredundancy check (CRC) code word), which is recovered by Turbo Codedecoding. After a predetermined number of iterations, the data portionof the decoded Turbo Codes is decoded by an appropriate error detectioncode decoder (e.g., a CRC decoder). Successful decoding by the errordetection code decoder provides an error detection.

The present invention is better understood upon consideration of thedetailed description below in conjunction with the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows model 100 of a communication system.

FIG. 2a shows schematically exemplary Turbo Code encoder 200.

FIG. 2b shows schematically exemplary Turbo Code decoder 250.

FIG. 3 shows the probability densities of bit errors per block forerroneous blocks after one iteration of Turbo Code decoding, in theresults of a simulation of a Gaussian channel using 2×10⁷ blocks ofTurbo Codes, with each block having a 110-bit message frame (i.e., notincluding the RSC codes), at a coding rate of ⅓ and a signal-to-noiseratio of 30 dB Hz.

FIG. 4 shows the distributions of indicator values in decoded blockswithout a flipped bit and in blocks found with one or more bits flipped,obtained based on the simulation of FIG. 3.

FIG. 5 shows probability density functions of a false negativedetermination (labeled 501) and of a false positive determination(labeled 502) for over the range of indicator threshold.

FIG. 6 shows a communication protocol that embeds an error detectioncapability in a transmitted message, while using a near-Shannon capacityTurbo Code encoding scheme, in accordance with one embodiment of thepresent invention.

FIG. 7 shows in table form the simulation results of communication usinga Turbo Codes encoding scheme, with error detection by CRC included,over a Gaussian channel.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Certain encoding scheme, such as Turbo Codes, do not have an errordetection capability. The present invention provides an error detectionability to a Turbo Code-based communication scheme.

According to one embodiment of the present invention, one may design anindicator that, when evaluated from the received modulated signal, canbe used to determine to some level of confidence that a transmittedsignal is correctly received. For example, suppose an analog value Y_(i)is received from modulated analog value X_(i) (which may be either −1.0or +1.0, inclusive) for the i-th bit of a transmitted message block. Themodulation may be, for example, a CDMA in conjunction with a phase shiftkey modulation scheme. For a Gaussian channel, the probability receivedsignal Y_(i) deviates from the mean of the transmitted value X_(i),P(Y_(i)-X_(i)|X_(i)), has a Gaussian distribution, which can beempirically obtained. Furthermore, for binary values transmitted over aGaussian channel, the probability that the i-th bit of a message blockof length L has the received value X_(i), as a result of the value beingtransmitted as X_(i) is given by:

$\frac{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)}{{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)} + {P\left( {Y_{i} - \overset{\_}{X_{i}}} \middle| \overset{\_}{X_{i}} \right)}},$

for i=1, 2, . . . L. This probability for each bit is independent fromthe same probability for any of the other bits in the message block,although they have the same probability distribution. This probabilitymay be interpreted to represent the probability that the i-th bit of themessage block, having value X_(i), is corrected received. Because oftheir independence, the log probability that all bits of the messageblock are correctly received is related to:

$\sum\limits_{i = 1}^{L}{\log \; \frac{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)}{{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)} + {P\left( {Y_{i} - \overset{\_}{X_{i}}} \middle| \overset{\_}{X_{i}} \right)}}}$

For a sufficient long message block (e.g., L≥40), this log probabilitymay be estimated by variable S with a Gaussian distribution:

$S = {\frac{- 1}{L}{\sum\limits_{i = 1}^{L}{\log \; \frac{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)}{{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)} + {P\left( {Y_{i} - \overset{\_}{X_{i}}} \middle| \overset{\_}{X_{i}} \right)}}}}}$

For a given message block, the amount the value of variable S relativeto its mean indicates the probability that one or more bits areincorrectly received. Thus, variable S may be used as an indicator thatcan serve as error detection.

In one embodiment of the present invention, in which the Turbo Codeencoding scheme of FIG. 2a is used, the sequence of received analogvalues {Y_(i)}, i=1, 2, . . . L, are digitized to the sequence of binaryvalues {{tilde over (X)}_(ι)}. Sequence {{tilde over (X)}_(ι)} is thenprovided to a Turbo Code decoder for decoding over a predeterminednumber of iterations (e.g., 1) to obtain decoded binary sequence {

}, in which

is the data portion of the Turbo Codes and

are the RSC codes. The binary data sequence {

} is then provided to a Turbo Code encoder to replicate the Turbo Codingthat occurred on the transmitter side prior to transmission. The TurboCode encoder provides the sequence {{circumflex over (X)}_(ι)}, which is{

}, where

are the computed RSC codes. {{circumflex over (X)}_(ι)} is then used tocompute variable S and its deviation from the mean value S provides anindication that the sequence {{circumflex over (X)}_(ι)} is a correctlyreceived message block.

FIG. 3 shows the probability densities of bit errors per block in blockswith one or more errors found after one iteration of Turbo Code decodingin the results of a simulation of a Gaussian channel using 2×10⁷ blocksof Turbo Codes, with each block having a 110-bit message frame (i.e.,not including the RSC codes), at a coding rate of ⅓ and asignal-to-noise ratio of 30 dB Hz. The simulation assumes a 888-chipCDMA code and a carrier frequency at 10⁸. FIG. 4 shows the distributionsof indicator values in decoded blocks without a flipped bit and inblocks found with one or more bits flipped, obtained based on thesimulation of FIG. 3. In FIG. 4, the distribution of indicator values inblocks without a bit flipped is labeled 401 and the distribution ofindicator values in blocks with one or more flipped bits is labeled 402.According to one embodiment of the present invention, a threshold valueof the indicator may be selected to decide whether a Turbo Code decodedvalue should be accepted as correct (“positive”) or not (“negative”),after a predetermined number of decoding iterations (e.g., oneiteration). For example, based on the FIG. 5, one may select a valuebetween 0.5 and 1.0, above which the Turbo Code decoding is accepted asincorrect and below which the Turbo Code decoding is accepted ascorrect.

The indicator threshold thus should be selected to simultaneouslycontain both false positives (i.e., accepting as incorrect value ascorrect) or false negative (i.e., rejecting a correct value asincorrect). FIG. 5 shows probability density functions of a falsenegative determination (labeled 501) and of a false positivedetermination (labeled 502) for over the range of indicator threshold.Using the data in FIG. 6, for example, if one selects an indicatorthreshold of 0.533, the probability of a false negative determinationfor a given block is less than 1%, while the probability of a falsepositive determination is 1.637%.

When a block is rejected as incorrectly received based on the indicator,a decoder may send a resend request. Alternatively, if the communicationprotocol sends an acknowledgement for an accepted block, anunacknowledged block is automatically resent after a predetermined timeperiod has elapsed.

According to one embodiment of the present invention and as illustratedin FIG. 6, prior to a block of data is encoded into Turbo Codes, thedata may first be encoded using a cyclic redundancy check (CRC) encodingscheme (step 601). The CRC-encoded data is then additionally encodedinto Turbo Codes (step 602). The Turbo Codes are then modulated andtransmitted over the communication channel (step 603). The receivedsignal is first demodulated (steep 604) and decoded in a Turbo Codedecoder (step 605). Each iteration of the Turbo Code decoder yields aCRC-encoded data portion and the RSC codes. The CRC-encoded data portionmay then be provided to a CRC decoder to recover the original data block(step 606). If the CRC-encoded data portion is successfully decoded, onemay conclude that no further Turbo Code decoding iteration is required.However, if the CRC-decoding is not successful, one may conclude that anerror may have occurred during transmission or that further iterationsof the Turbo Code decoding may be required. At that point, one optionmay be to request a retransmit from the transmitting side, or to carryout one or more iterations of Turbo Code decoding to see the resultingCRC-encoded data portion can be successfully CRC-decoded.

FIG. 7 shows in table form the simulation results of communication usinga Turbo Codes encoding scheme, with error detection by CRC included,over a Gaussian channel. As in the simulation of FIG. 3, the simulationin FIG. 7 uses 2×10⁷ blocks of Turbo Codes, with each block having 110bits compose of a 100-bit message frame and CRC parity bits, encoded ata coding rate of ⅓ and a signal-to-noise ratio of 30 dB Hz. In thisinstance, the CRC-encoding has a generator polynomial given by:

g(x)=x ¹⁰ +x ⁹ +x ⁵ +x ⁴ +x+1

FIG. 7 lists in row order (i) the number of errors in the 2×10⁷ blocksof Turbo Codes, (ii) the corresponding bit error rate, (iii) the numberof errors detected by CRC decoding, and (iv) the error rate achieved byaccepting a successful CRC-decoding, after each of Turbo Code decodingiterations 1-18 As shown in FIG. 7, 34035 blocks are shown to containerror bits after one iteration of Turbo Code decoding, thereby achievinga bit error rate of 1.70175×10⁻³ bit error rate. However, 34024 of theseerrors are uncovered by CRC decoding after one iteration of Turbo Codedecoding. Thus, the simulation in FIG. 7 shows that embedding the errordetection capability of a CRC in Turbo Codes achieves in the example abit error rate of 5.5×10⁻⁷ after just one iteration of Turbo Codedecoding. FIG. 7 shows that, if error detection by CRC is not used, theequivalent error rate is not achieve until after 10 iterations of TurboCode decoding. Thus, embedding CRC error detection in Turbo Codes savesin this instance at least 9 iterations of Turbo Code decoding.

In fact, by combining error detection using the indicator with errordetection using CRC codes, as taught above, a bit error rate of9.0035×10⁻⁹ is achieved after one Turbo Code decoding iteration.

As one can see from the above, by augmenting Turbo Codes with an errordetection capability, the number of iterations necessarily for decodingTurbo Codes can be significantly reduced, thereby shortening decodinglatency by six-folds or more in many practical applications.

The above detailed description is provided to illustrate the specificembodiments of the present invention and is not intended to be limiting.Numerous modification and variations of the present invention ispossible within the scope of the present invention. The presentinvention is set forth in the following claims.

1. A method for receiving a message encoded in Turbo Codes and modulatedfor transmission as an analog signal, comprising: demodulating theanalog signal to recover the Turbo Codes; and decoding the Turbo Codesto recover the message using an iterative Turbo Code decoder, whereinthe decoding performs an error detection after a predetermined number ofiterations of the Turbo Code decoder to determine whether or not anerror has occurred during the transmission.
 2. The method of claim 1,wherein the predetermined number of iterations is one.
 3. The method ofclaim 1, wherein decoding stops when the error detection determines thatan error has not occurred during the transmission.
 4. The method ofclaim 1, wherein a request for retransmission of the message is sentwhen the error detection determines that an error has occurred duringthe transmission.
 5. The method of claim 4, wherein a further iterationof decoding is carried out in the Turbo Code decoder when the errordetection determines that an error has occurred during the transmission.6. The method of claim 1, wherein the error detection step comprisesevaluating an indicator having a value that is related to a probabilitythat an error has occurred during the transmission.
 7. The method ofclaim 6, wherein the analog signal is transmitted at one of a pluralityof analog values, and wherein the probability is based on the receivedanalog signal's deviation from the transmitted analog signal.
 8. Themethod of claim 7, wherein the probability relates toP(Y_(i)-X_(i)|X_(i)), where X_(i) and Y_(i) represent an analog value ofthe transmitted analog signal and a value of the received analog value,respectively.
 9. The method of claim 8, wherein evaluating the indicatoris includes computing:$\sum\limits_{i = 1}^{L}{\log \; \frac{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)}{{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)} + {P\left( {Y_{i} - \overset{\_}{X_{i}}} \middle| \overset{\_}{X_{i}} \right)}}}$where L is a length in number of bits of the Turbo Codes and X_(i) is ananalog value representing a complement of X_(i).
 10. The method of claim8, wherein the recovered message is used in a Turbo Code encoder toregenerate a regenerated Turbo Code word, and wherein the probability iscomputed based on the regenerated Turbo Codes.
 11. The method of claim6, wherein the message comprises an error detection code, and whereinthe error detection comprises recovering the error detection code. 12.The method of claim 1, wherein the message includes an error detectioncode, and wherein the error detection comprises decoding the errordetection code.
 13. The method of claim 12, wherein the error detectioncomprises determining that a decoding of the error detection code issuccessful.
 14. The method of claim 12, wherein the error detection codeis based on cyclic redundancy check.
 15. The method of claim 14, whereinthe cyclic redundancy check is based on a generator polynomial given by:g(x)=x ¹⁰ +x ⁹ +x ⁵ +x ⁴ +x+1
 16. A decoder for receiving a messageencoded in Turbo Codes and modulated for transmission as an analogsignal, comprising: a demodulator that demodulates the analog signal torecover the Turbo Codes; and an iterative Turbo Code decoder thatdecodes the Turbo Codes to recover the message, wherein the decodingincludes performing an error detection after a predetermined number ofiterations of the Turbo Code decoder to determine whether or not anerror has occurred during the transmission.
 17. The decoder of claim 16,wherein the predetermined number of iterations is one.
 18. The decoderof claim 16, wherein decoding stops when the error detection determinesthat an error has not occurred during the transmission.
 19. The decoderof claim 16, wherein a request for retransmission of the message is sentwhen the error detection determines that an error has occurred duringthe transmission.
 20. The decoder of claim 19, wherein a furtheriteration of decoding is carried out in the Turbo Code decoder when theerror detection determines that an error has occurred during thetransmission.
 21. The decoder of claim 16, wherein the error detectioncomprises evaluating an indicator having a value that is related to aprobability that an error has occurred during the transmission.
 22. Thedecoder of claim 21, wherein the analog signal is transmitted at one ofa plurality of analog values, and wherein the probability is based onthe received analog signal's deviation from the transmitted analogsignal.
 23. The decoder of claim 22, wherein the probability relates toP(Y_(i)-X_(i)|X_(i)), where X_(i) and Y_(i) represent an analog value ofthe transmitted analog signal and a value of the received analog value,respectively.
 24. The decoder of claim 23, wherein evaluating theindicator is includes computing:$\sum\limits_{i = 1}^{L}{\log \; \frac{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)}{{P\left( {Y_{i} - X_{i}} \middle| X_{i} \right)} + {P\left( {Y_{i} - \overset{\_}{X_{i}}} \middle| \overset{\_}{X_{i}} \right)}}}$where L is a length in number of bits of the Turbo Codes and X_(i) is ananalog value representing a complement of X_(i).
 25. The decoder ofclaim 23, wherein the recovered message is used in a Turbo Code encoderto regenerate a regenerated Turbo Code word, and wherein the probabilityis computed based on the regenerated Turbo Codes.
 26. The decoder ofclaim 21, wherein the message comprises an error detection code, andwherein the error detection comprises recovering the error detectioncode.
 27. The decoder of claim 16, wherein the message includes an errordetection code, and wherein the error detection comprises decoding theerror detection code.
 28. The decoder of claim 27, wherein the errordetection comprises determining that a decoding of the error detectioncode is successful.
 29. The decoder of claim 27, wherein the errordetection code is based on cyclic redundancy check.
 30. The decoder ofclaim 29, wherein the cyclic redundancy check is based on a generatorpolynomial given by:g(x)=x ¹⁰ +x ⁹ +x ⁵ +x ⁴ +x+1